Large deviation inequalities for supermartingales and applications to directed polymers in a random environment

Quansheng Liu; Frédérique Watbled

doi:10.1016/j.crma.2009.08.004

Probability Theory

Large deviation inequalities for supermartingales and applications to directed polymers in a random environment
[Inégalités de grandes déviations pour les surmartingales et applications aux polymères dirigés en milieu aléatoire]

Présenté par : Jean-Pierre Kahane

Quansheng Liu ^1,² ; Frédérique Watbled ²

¹ Changsha University of Science and Technology, School of Mathematics and Computing Science, Changsha, 410076, PR China
² Université de Bretagne-Sud, LMAM, campus de Tohannic, BP 573, 56017 Vannes, Université Européenne de Bretagne, France

Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1207-1212.

Résumé
Abstract

Nous montrons d'abord des inégalités exponentielles de grandes déviations pour les surmartingales, qui sont optimales dans le cas de variables aléatoires indépendantes et identiquement distribuées (iid). Nous utilisons ensuite ce résultat pour établir des inégalités exponentielles de concentration pour l'énergie libre des polymères dirigés en milieu aléatoire ; nous en déduisons des bornes supérieures pour sa vitesse de convergence (en probabilité, presque sûrement, et dans $L^{p}$ ), et une expression de l'énergie libre en terme d'énergies libres de certaines cascades multiplicatives.

We first show exponential large deviation inequalities for supermartingales, which are optimal in the independent and identically distributed (iid) case. We then apply them to establish new exponential concentration inequalities for the free energy of directed polymers in random environment; as consequences we obtain upper bounds of its rate of convergences (in probability, almost surely, and in $L^{p}$ ), and give an expression for the free energy in terms of free energies of some multiplicative cascades.

Reçu le : 2009-05-06
Accepté le : 2009-08-07
Publié le : 2009-09-05

DOI : 10.1016/j.crma.2009.08.004

Affiliations des auteurs :

Quansheng Liu ^{1, 2} ; Frédérique Watbled ²

¹ Changsha University of Science and Technology, School of Mathematics and Computing Science, Changsha, 410076, PR China
² Université de Bretagne-Sud, LMAM, campus de Tohannic, BP 573, 56017 Vannes, Université Européenne de Bretagne, France

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     author = {Quansheng Liu and Fr\'ed\'erique Watbled},
     title = {Large deviation inequalities for supermartingales and applications to directed polymers in a random environment},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1207--1212},
     publisher = {Elsevier},
     volume = {347},
     number = {19-20},
     year = {2009},
     doi = {10.1016/j.crma.2009.08.004},
     language = {en},
}

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Quansheng Liu; Frédérique Watbled. Large deviation inequalities for supermartingales and applications to directed polymers in a random environment. Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1207-1212. doi : 10.1016/j.crma.2009.08.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.08.004/

Bibliographie
Cité par

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[5] Q. Liu On generalized multiplicative cascades, Stochastic Process. Appl., Volume 86 (2000) no. 2, pp. 263-286

[6] Q. Liu, F. Watbled, Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment, Stochastic Process. Appl., in press

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